Theorem imaeq2d 4692

Description: Equality theorem for image. (Contributed by FL, 15-Dec-2006.)

Hypothesis

Ref	Expression
imaeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
imaeq2d	|- ( ph -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2d

Step	Hyp	Ref	Expression
1		imaeq1d.1	. 2 |- ( ph -> A = B )
2		imaeq2 4688	. 2 |- ( A = B -> ( C " A ) = ( C " B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1285   "cima 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-cnv 4373  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378

This theorem is referenced by:  imaeq12d  4693  nfimad  4701  elimasng  4717  ressn  4882  foima  5136  f1imacnv  5168  fvco2  5268  fsn2  5363  resfunexg  5408  funfvima3  5418  funiunfvdm  5428  isoselem  5484  fnexALT  5765  eceq1  6200  uniqs2  6225  ecinxp  6240  phplem4  6380  phplem4dom  6387  phplem4on  6392